First principles global optimization of metal clustersand nanoalloysJaeger, Marc; Schaefer, Rolf; Johnston, Roy

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First principles global optimization of metalclusters and nanoalloys

Marc Jäger, Rolf Schäfer & Roy L. Johnston

To cite this article: Marc Jäger, Rolf Schäfer & Roy L. Johnston (2018) First principles globaloptimization of metal clusters and nanoalloys, Advances in Physics: X, 3:1, S100009, DOI:10.1080/23746149.2018.1516514

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First principles global optimization of metal clusters andnanoalloysMarc Jägera, Rolf Schäfera and Roy L. Johnstonb

aEduard-Zint-Institut für Physikalische und Anorganische Chemie, Technische Universität Darmstadt,Darmstadt, Germany; bSchool of Chemistry, University of Birmingham, Birmingham, UnitedKingdom

ABSTRACTThe global optimization of nanoparticles, such as pure orbimetallic metal clusters, has become a very important andsophisticated research field in modern nanoscience. Thepossibility of using more rigorous quantum chemical firstprinciple methods during the global optimization has beenfacilitated by the development of more powerful computerhardware as well as more efficient algorithms. In this review,recent advances in first principle global optimization meth-ods are described, with the main focus on genetic algo-rithms coupled with density functional theory foroptimizing sub-nanometre metal clusters and nanoalloys.

ARTICLE HISTORYReceived 2 July 2018Accepted 16 August 2018

KEYWORDSFirst Principles; ab initio;global optimization; metalclusters; nanoparticles;nanoalloys; geneticalgorithms; electronicstructure methods; globalminimum; quantumchemistry; density functionaltheory; DFT

CONTACT Marc Jäger jaeger@cluster.pc.chemie.tu-darmstadt.de Eduard-Zint-Institut für Physikalischeund Anorganische Chemie, Technische Universität Darmstadt, Darmstadt, Germany

ADVANCES IN PHYSICS: X2018, VOL. 3, NO. 1, 1516514https://doi.org/10.1080/23746149.2018.1516514

© 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Introduction

In today’s world, the global optimization (GO) of functions according toone or more criteria has become very popular in many different researchfields and applications [1–10]. By the term ‘global optimization’ we meanfinding the overall best solution for a given mathematically formulatedproblem, which usually corresponds to the global maximum or the globalminimum (GM) of a function or a set of functions.

An important goal in modern nanoscience is the control of materials onthe atomic scale, coupled with achieving a fundamental understanding ofhow physicochemical properties depend on the particle structure in orderto tailor nanomaterials for real-world applications [11]. Thus, computer-aided optimization can be used to predict a suitable atomic structurecorresponding to an ideal (or near-ideal) value of the property of interestand to interpret experimental results, as well as guiding the development offuture materials in the course of rational material design. Therefore, GO ofnanosystems plays a very important role.

Metal clusters have received enormous interest due to their industrialapplications in nanotechnology, electronical, biological, medical devicesand catalysis [11–35]. Furthermore, metal clusters are ideal test systemsfor probing physical theories. The limits of concepts for bonding theoriescan be evaluated particularly well with the help of high-quality experi-mental data. The range of properties of metallic clusters can be signifi-cantly enhanced by combining more than one metal, forming ‘nanoalloys’[14]. These mixed metal clusters allow the modification of chemical andphysical properties by changing, in addition to their size and shape, theircomposition and chemical ordering. The system complexity increasessignificantly for mixed clusters due to the presence of ‘homotops’ (inequi-valent permutational isomers), which makes the GO a more challengingtask. For nanoalloys, there are also several possible mixing patterns (che-mical ordering). Common types of chemical ordering are: random orordered mixed; core-shell and multi-layer (onion-like); and phase segre-gated (Janus) [14]. The preferred chemical ordering depends on severalfactors and is a balance between the cohesive energies, surface energies,relative atomic sizes and specific electronic and magnetic effects.

The possibility of performing GO on ever larger systems and applying morerigorous quantum chemical electronic structure methods has gone hand-in-hand with the development of increasing computing power. For an accuratedescription of the smallest clusters (or sub-nanometre particles), a quantumchemical description is essential, due to the increasing importance of quantumsize effects. The most rigorous methods are ab initio electronic structuremethods, which only rely on the laws of nature without making use of empiri-cally fitting parameters or additional assumptions [36]. The input to ab initio

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electronic structure methods are physical constants, atomic numbers, the posi-tions of the nuclei and the number of electrons [37]. Popular ab initiomethodsare Hartree-Fock (HF) methods and post-HF methods such as Møller-Plesset-Perturbation theory (MPn), Configuration Interaction (CI) and CoupledCluster (CC) methods. Another very widely used method in material scienceis Density Functional Theory (DFT). Though DFT is strictly not an ab initioelectronic structure method, in this review we will describe both DFT and abinitio electronic structure methods by the term ‘first principles’, as in the reviewby Heiles and Johnston [10].

The main focus in this review is to introduce and present examples of GOmethods, applied to sub-nanometre pure metal clusters and nanoalloys, usingfirst principle methods.

Global optimization methods for metal clusters and nanoalloys

Within the Born Oppenheimer approximation, stable structures for a givencluster (e.g. different stable isomers) correspond to local minima on themultidimensional potential energy surface (PES). The PES of a clusterdescribes its energy as a function of its atomic coordinates. GO corre-sponds to finding the most stable atomic arrangement for a specific cluster(defined by the number and type of its constituent atoms as well as by thetotal charge) that is the lowest energy-point on the PES, the so-called‘global minimum’. The significant effort that has been expended in findingcluster GM [10,38–48] is because the GM is usually the most likelystructure to be formed in an experiment (though kinetic factors shouldnot be forgotten). Even if the experimental structure is not the GM,typically it is a very low energy isomer [46]. In combined experimentaland theoretical studies, the presence of one or more low lying isomers havebeen found to explain experimental results very well [49–51]. Theapproach in such combined studies is: (i) to measure a physico-chemicalproperty of a particular cluster; (ii) to use a GO technique to search for theGM and other low energy structures and (iii) to calculate the sameproperty for these isomers. Finally, the comparison between theory andexperiment can lead to structural assignment discrimination of the cluster(see Figure 1) [51,52]. However, this approach is complicated by the factthat the number of possible stable structures (local minima on the PES)increases (at least) exponentially with the size of the cluster and, hence, sodoes the computational effort required. For mixed clusters, such as nanoal-loys, the problem is compounded due to the combinatorial increase ofpossible minima (so-called ‘homotops’ [29]) resulting from swappingpositions of different elements.

Deterministic location of the GM would be possible with an exactknowledge of the PES. However, this requires calculating the entire PES

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Figure 1. Combined experimental and theoretical approach for cluster structure discrimina-tion. (a) Reproduced from Ref. [52] with permission from Wiley-VCH Verlag GmbH & Co:compares experimental electrostatic beam deflection measurements and calculated beamprofiles for the globally optimized Sn6Bi3 cluster and other low-lying isomers. (b) Reproducedfrom Ref. [51] with the permission of AIP Publishing: The experimental absorption spectraobtained by UV-Vis photodissociation spectroscopy is shown, along with the calculatedabsorption spectra for the lowest lying AgAu3

+ isomers.

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to determine all its minima and saddle points, which is only possible forthe simplest cases and is generally not feasible using relatively expensive abinitio (or even DFT) electronic structure methods, for all but the smallestsystems. For these reasons, sophisticated GO methods are essential toperform an efficient and unbiased exploration of the PES. In general, agood GO strategy must combine both local and global aspects of PESsearching. Whereas local searching addresses the sampling of single solu-tions, global searching deals with the efficient exploration of differentregions of the multidimensional parameter space [53].

In constructing a GO for clusters, there are two issues to deal with.From a mathematical point of view this corresponds to the questions ofhow to choose good discrete argument values and how to calculate thetarget set. The first question pertains to the GO algorithm for generatingcluster structures (defining atomic coordinates) in order to efficientlyexplore different regions on the PES. The second question concerns thelevel of theory used to calculate the energies (points on the PES) for thegenerated geometries.

Common classes of algorithm which are used for the GO of clusters are:Genetic algorithms (GA) [46]; basin hopping (BH) [45]; particle swarmoptimization (PSO) [54]; artificial bee colony (ABC) [55]; simulatedannealing [56] and threshold algorithms [57]. For a given cluster, with afixed number of atoms and elemental composition, the computational timestrongly depends on the selected level of theory. Therefore, a trade-offbetween accuracy and cost must be found. The choice of the appropriatelevel of theory for describing the chemical bonding between the atoms ofthe cluster depends on the system size as well as the available computa-tional resources, which are dependent on the development of more effi-cient and ever faster computer hardware. Hence, in early GO studies ofclusters, generally only empirical (or semi-empirical) potentials (EP) wereused [58–66]. Even today, many studies employ this EP-GO approach,which is usually followed by geometry refinement with more rigorous firstprinciple methods [67–79]. The main advantage of this approach is that itcan also be applied to large clusters, of several hundreds or a few thou-sands of atoms. For larger nanoparticles (up to many tens or hundreds ofatoms), which already show similar properties and chemical bonding as inthe bulk phase (with properties typically scaling as the surface to volumeratio, i.e. as 1/R), the EP-GO approach yields reasonable results [38,58,80-87]. Furthermore, EP-GO is not only beneficial when systems already havebulk-like behaviour, but also when larger clusters form unusual structuralmotifs so that it would be difficult to guess by chemical intuition alone. Forinstance, chiral icosahedral and decahedral nanoalloys in the size rangebetween 100 and 200 atoms as well as pyritohedral structures and chiraldecahedra for nanoalloys up to 586 atoms have been found with the EP-

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GO approach and subsequent first principle calculations [71,73].Unfortunately in general, simple EPs do not reproduce subtle effects dueto the intrinsic quantum chemical nature of chemical bonding, whichbecome increasingly important with decreasing system size, i.e. in theregion where ‘every atom counts’ [88]. With the term ‘intrinsic quantumchemical nature of chemical bonding’, we mean that in EPs no electronicinteractions (as well as no explicit electronic correlation effects) are con-sidered, in contrast to first principle methods [89,90]. For example, Lyonset al. examined with an EP the relative energetics of C44 isomers [91]. Theyfound an overall good agreement with DFT results, but they emphasizedthat electronic effects are not considered in the EP calculations and specialstability due to resonance effects is not addressed in their study. Althoughin principle it is possible to build an EP for a proper description of somespecific small clusters, such potentials may not be transferable to slightlylarger clusters. There are several examples where first principle calculationsand experimental findings disagree with EP computations regarding thecluster structure or relative energies of different isomers. For example, GOof trimetallic AgkAumPtn (k + m + n = 13, 19, 33, 38) clusters result indifferent GMs for EP and DFT calculations [92]. GO of Sin (n = 16–20)clusters employing three different EPs, as well as DFT computations,provides different GMs for each theory level [64]. A combined EP andDFT study of 40-atom Pt-Au clusters shows only a partial agreementbetween EP and DFT results [80]. Moreover, a theoretical EP study ofthe structures of neutral Agn clusters (n = 2–60) suggests a decahedralcluster growth and compared this result with experimental trapped ionelectron (TIED) findings of silver cations of the same size, but the experi-mental data suggest an icosahedral growth [93]. Hence, in general, firstprinciples electronic structure methods are the better choice for the accu-rate description of the chemical bonding (and hence for GO) of small sub-nanometre clusters [89]. When only dealing with the optimization ofchemical ordering within the cluster using EP-based optimization, thepresent size limits are well above 1000 atoms for an unconstrained searchand several thousand atoms for symmetry constrained searches [75,94].Regarding merely the optimization of chemical ordering, a DFT optimiza-tion of a 309-atom AuCu cluster (an icosahedron) has been realized [74].Another method that has been applied is the semi empirical tight-bindingDFT (TBDFT) method. The TBDFT approach has empirical parameters,but can maintain close-to-DFT accuracy, for considerably reduced compu-tational cost [95]. Hence, provided it is used within the region of para-meterization, DFTB can be an efficient method for metal cluster GO,which was recently shown for TBDFT applications to silver and goldclusters [96]. A study has also been reported which combines TBDFTand DFT directly in the GO of gold clusters [97].

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Usually, when performing first principles GO of clusters and nanopar-ticles, the GO algorithm is coupled to a quantum chemical (QC) softwarepackage. The choice of the particular electronic structure method andpackage depends on available computer resources and on the size andcomplexity of the cluster system (e.g. pure metal cluster vs. nanoalloy,presence or absence of ligands and/or supporting surface, importance ofmagnetism).

GO algorithms can be distinguished in a number of ways [10]. Firstly, theycan be divided into biased and unbiased methods. In biased GO, prior knowl-edge (or guesses) about bonding in the system or preferred structural motifs areused to accelerate the exploration of the PES. In a strictly unbiased global search,however, no prior knowledge is used. Another classification concerns theenergy calculation for a generated geometry. During the GO process, thereare in general two possibilities for how to perform the local energy calculation:Either only the energy of the generated cluster geometry is calculated (singlepoint calculation) or for each created cluster structure a local geometry optimi-zation (energy minimization) takes place so that each cluster structure corre-sponds to a local energy minimum on the PES.

In the optimization of clusters, a cost-saving approach (which some-times gives good results) involves first performing GO at a lower level oftheory (EP or DFT with a small basis set) and looser convergence criteria.This is followed by refinement (reoptimization) of a set of low-energy andstructurally distinct isomers using a more rigorous ab initio method, suchas CCSD(T), or DFT with larger basis sets and/or more computationallyexpensive hybrid exchange-correlation (xc) functionals [10,50,51,98–103].Particular attention must be paid to the fact that the relative energydifferences between the isomers, as well as their energetic order, canchange (especially in DFT with the use of different xc functionals andbasis sets) so that an originally energetically higher isomer can become theGM after refinement (see Figure 2) [51]. It is also possible that the hightheory level GM is missed if the lower level PES is not a sufficiently goodmatch to the ‘true’ PES in the low energy regions.

In the following sections, while several GO methods will be brieflymentioned, we will focus on unbiased GO of metal clusters and nanoalloysat the DFT level, principally using GA.

Genetic algorithms for optimizing cluster structures

The GA is a metaheuristic based on the principles of natural evolution andbelongs to the class of evolutionary algorithms (EA), which are examples ofthe emerging area of artificial intelligence (AI). The GA can be seen as anintelligent search algorithm, which learns features of good solutions by therecognition of schemata or building blocks during the exploration of the

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multidimensional parameter (coordinate) space. The GA successivelyimproves its solution during the search progress. We only give a briefdescription of the GA procedure here and refer the reader to the literaturefor more detailed explanations [46,104].

In general, GAs can be applied to any GO problem where the variables(genes) can be encoded as a string (chromosome), which represents a testsolution of the problem. The overall task is the GO of the values of thevariables (‘alleles’) in order to find the overall best solution [46]. To dothis, the GA uses evolutionary processes such as mutation, mating (alsoknown as crossover) and natural selection. The GA starts with an initialpopulation of a number of individuals (with the population size generallybeing inversely related to the computational cost of the energy minimiza-tion step), which are usually generated randomly [10]. During the GAprocess, the population evolves for a specified number of generations oruntil energy or population convergence is reached, by applying the afore-mentioned evolutionary operators (crossover and mutation) to each gen-eration. The crossover method mimics biological genetic crossover,combining genetic information from two parent strings (in a GA, morethan two parents can be used), in order to produce one (or more) off-spring. The most common crossover operators are one-point and two-point crossover. In one-point crossover, the parent strings are cut at thesame position and offspring are generated by adding complementary

Figure 2. Lowest energy isomers for Ag2Au2+ for different xc functionals for the same def2-

TZVPP basis set. Different xc functionals lead to different relative energies and may also causea different energy order of the isomers. Reproduced from Ref. [51], with the permission of AIPPublishing.

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(parent) genes. In two-point crossover, the parent strings are cut at twodifferent positions. To ensure a sufficient diversification of genetic infor-mation for each generation, a mutation operator is used which suppliesnew genetic material. This is to prevent stagnation of a population, corre-sponding to convergence on a non-optimal solution.

In each generation, the fitness (objective function in the GO problem) ofevery individual is evaluated and serves as a measure of the quality of thetrial solution. The more fit individual members of the current populationare chosen for subsequent crossover or mutation in order to breed a newgeneration. For this purpose, popular selection methods are roulette wheeland tournament selection. This selection process imitates the naturalselection concept in biological evolution, which is also known as ‘survivalof the fittest’. Consequently, it is an elitist strategy, because the best(highest fitness) individual is guaranteed to survive into the subsequentgeneration, thereby enhancing the probability of passing on some of its‘good’ genetic material. Thus, a gradual improvement of the trial solutionswill be achieved by applying GAs to GO problems. There are additionalstrategies which have been introduced to improve the performance of theGA. One strategy which can help to accelerate the GM search is biasing theGA by introducing a seeded initial population [105]. Another strategy is toensure a certain degree of diversity in each population in order to avoidstagnation (that is to avoid being trapped in a sub-optimal local minima).This can be accomplished with replacement or deletion (also known aspredator) strategies. These strategies define which member(s) of the popu-lation will be replaced with a new one (in addition to the previouslymentioned fitness-based replacement) or deleted. With these methods itis possible to remove identical (or at least too similar) members in onepopulation in order to increase the diversity. Those strategies require oneor several dissimilarity measures via the definition of descriptor operatorsto compare each member of the population. The results are usually a set ofnumbers or arrays of dissimilarity measures indicating how the individualsdiffer from each other. Thereafter a ‘predator operator’ removes certainmembers of the population, those which are too similar or identicalaccording to the chosen descriptors [105-112]. In the case of clusters,commonly used descriptors for dissimilarity measures are: energies,moments of inertia, radial distribution functions, connection tables, var-iance of atomic distances, etc. [105,106,109-112].

The first applications of GAs for finding the GM structure of clustersdates back to 1993 for atomic clusters (Si4) by Hartke [41] and molecularclusters (benzene dimer, trimer and tetramer) by Xiao and Williams [113].In these first publications, the cluster coordinates were encoded as binarystrings and thus represented as a chromosome to which the evolutionaryoperators (crossover and mutation) were applied. In 1995, Zeiri introduced

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a GA which directly used real number cartesian coordinates of the clusteratoms and hence eliminated the representation problem (encoding anddecoding the cluster coordinates to binary strings) [43]. The next mile-stone in GA development was due to Deaven and Ho who introduced aphysical cut-and-splice method for the crossover operator, as depicted inFigure 3 [42]. This operation is highly effective for the GO of clusters sinceit considers the spatial distribution of the atoms and hence is sensitive tostructural properties (this means it is able to pass ‘good’ structural featuresfrom the chosen parents to the new child cluster generation) as well asbeing highly effective. Thus, the cut and splice operator is widely used andimplemented in many GAs for clusters and nanoparticles. The cut andsplice operation (see Figure 3) applies random rotations of both parents,uses a horizontal cutting plane, cuts each parent and combines comple-mentary parts of each of one of the parents to create descent(s). Morerecent studies have developed further crossover operators for clusters[114–116]. One new form is the generalized cut and splice (GenC&S)operator [114,115]. In contrast to the Deaven-Ho approach, the GenC&Soperator does not make use of a plane or random rotations of the parentsbefore mixing the genetic material. It uses the Euclidean distance as thecriterion for selecting atoms of each cluster such that subsets of atoms thatare close together in the parent clusters will form the new building blocksto create the offspring. In the GenC&S version there is no bias associatedin the operator as in the Deaven-Ho approach (that is using a horizontalcutting plane and forcing the new sub-cluster to be parallel to this cuttingplane) [115]. Chen et al. pointed out that the cut and splice operator ofDeaven and Ho may fail in the case where the cutting plane lies near themajor axis of one parent and near the minor axis of the other parent [116].In this case, the resulting structure is unstable and after local minimizationthe final structure shows no apparent relationship to the parents. Hence,good (low energy) structural features of the parents would not be passed tothe offspring. For this reason, Chen et al. proposed a new so-called ‘sphere-cut-splice’ crossover which employs a sphere instead of a cutting plane andgenerates the offspring by using atoms of one parent, which lie inside thesphere and atoms of the other parent, which lie outside the sphere. Thismethod is able to preserve good schemata and has been shown revealed tobe very suitable for the GO of larger clusters.

Another innovation of Deaven and Ho was the introduction of localenergy optimizations for each generated cluster structure, so that eachgenerated cluster is relaxed to the nearest local minimum. This approachwas initially applied to fullerenes (C20 to C60) [42]. GAs which couplelocal minimizations to the global search are called ‘Lamarckian’ because,from a biological point of view, this process corresponds to Lamarckianrather than Darwinian evolution [46]. Doyle and Wales demonstrated

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that local minimizations effectively transform the PES into a steppedsurface in which each step corresponds to a basin for the respective localminimum [62]. The Lamarckian approach, thereby, improves the effi-ciency of the GA tremendously.

Many different GAs have been developed for the GO of clusters and appliedto a myriad of cluster systems [8,10,38-42,46,58,81,98,104,105,111,113,117-123,124-131].

The GA-DFT approach

A widely used first principles approach for GO of clusters at the DFT levelis the GA-DFT approach. The first GA-DFT approach was introduced in1996 by Tomasulo and Ramakrishna who performed GO of AlP-clustersdirectly at the DFT level of theory, within the local density approximation[117]. Several years later, Alexandrova and Boldyrev firstly coupled a GAcode to a quantum chemical software package and introduced the gradientembedded genetic algorithm (GEGA) [132]. Other recently developed firstprinciples based GAs are, for example, OGOLEM [124], a versatile pool-based GA implementation for clusters and flexible molecules and a GAthat uses machine-learning techniques to improve its performance [129].Though the GA-DFT approach is the most widely used method for the GOwith first principle methods, there is also a GA study which uses MP2calculations for sodium potassium nanoalloy clusters [133]. The followingbriefly describes the development of the GA-DFT approach within the

Figure 3. Representation of the Deaven and Ho cut-and-splice crossover operation.Reproduced from Ref. [46] with permission from The Royal Society of Chemistry.

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Birmingham Cluster Genetic Algorithm and its derivatives, developed byJohnston and collaborators.

The Birmingham Cluster Genetic Algorithm (BCGA)

The BCGA is a Lamarckian-GA, which is capable of performing GO ofbare pure or mixed clusters at either the EP or DFT level of theory [46,98].In this algorithm, clusters are represented as real-valued cartesian coordi-nates (as introduced by Zeiri) on which the genetic operators act directly.In general, such GAs are called ‘phenotype algorithms’ [111]. The Deaven-Ho cut-and-splice crossover method and several mutation methods (atomdisplacement, twisting, cluster replacement and (inequivalent) atom per-mutation) are employed. In order to perform GO using plane wave (PW)DFT, the BCGA was initially coupled to the software package QuantumEspresso. The first study with this BCGA-DFT approach was the investiga-tion of the dopant-induced 2D–3D transition of 8-atom Au-Ag nanoal-loys [98].

A major disadvantage of the traditional GA approach is that the localenergy minimization steps are performed sequentially, which results in abottleneck in the GO procedure and limits the cluster sizes which can bereliably optimized. This is especially important when using first principlesmethods in the local minimizations. For example, employing DFT calcula-tions for GO, typically more than 99.9% of the computational time is spentin the local minimization steps [124,130]. Therefore, parallelization of theGA, taking advantage of larger computational resources, became absolutelynecessary.

The Pool-BCGA code was developed [39] to allow several independent localoptimizations to be performed at the same time during the GO. In the poolapproach, multiple instances of the GA act on a global database (pool) in orderto perform simultaneous local optimizations of cluster structures which aregenerated by crossover and mutation. These GA subprocesses share the GOworkload (as depicted in Figure 4), which leads to parallelization of the algo-rithm. During GO, the pool database consists of the geometric and energeticinformation of a fixed number of isomers. Each individual GA instance uses thepool by applying crossover and mutation operators to the database members inorder to generate new structure candidates. Each newly formed isomer com-petes with the current members of the pool and replaces the highest energydatabase structure if the new individual has a lower energy (higher fitness). Atthe beginning of the GO, an initial pool is created by random generation andlocal minimization of a pre-determined number of structures. After the data-base is filled, the pool-clusters are randomly selected according to their fitnessusing the roulette wheel method and are subjected to the crossover and muta-tion operators. The pool-GA was benchmarked and applied successfully to the

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GO of Au10 and Au20 clusters. An evolution plot of the GO of Au10 is shown inFigure 5. In addition, the influence of the mutation rate on the evolutionaryprocess was investigated for Au10Pd10 and it was found that the higher themutation rate the greater the number of higher-energy-isomers that were foundand the more structures had to be generated to find the GM. This exampleshows that too high mutation rates reduce the efficiency of finding the GM.

The Fortran-based pool-BPGA was subsequently incorporated within amore efficient modern Python framework and was coupled to the Vienna Abinitio Simulation Package (VASP), which is a plane wave DFT code with PAWpseudo-potentials [134-137]. The resulting Birmingham Parallel GeneticAlgorithm (BPGA) was first applied to the GO of iridium clusters with 10 to20 atoms. The methodology of the BPGA is shown in Figure 6 [40]. The BPGAcan also be used for the optimization of cluster geometries in the presence of asurface and applied to surface-grown or soft-deposited clusters in order to studycatalytic active supported clusters [138,139].

The latest published GA-DFT code based on the BPGA is the MexicanEnhanced Genetic Algorithm (MEGA), written by Vargas, Beltran and co-workers [130]. The code is also written in Python and coupled to the VASPcode. MEGA is more efficient and flexible and a number of new features havebeen introduced, such as progress documentation (a text file, which documentsthe applied evolutionary operator to each generated cluster structure) andseveral new mutation implementations, such as ‘move’ or ‘twist’. The ‘move’mutation performs a random displacement of 25% of the atoms, while ‘twist’mutation rotates one half of the cluster atom by a random angle with respect to

Figure 4. Scheme of the pool-GA concept. Several GA subprocesses work on the pooldatabase in parallel. Reproduced from Ref. [39] with permission from PCCP Owner Society.

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the other half. This algorithm can start from scratch with randomly generatedstructures for performing an unbiased GO, but it can also operate in biasedmode, starting from predefined pool structures (‘seeded pool’). One ofMEGA’smain advantages is the use of two criteria (energetic and structural comparison)for testing isomer diversity, deleting similar structures and adding new struc-tures as required. Hence, the structural diversity of the pool is maintainedduring the whole GO, which decreases the probability of stagnation and

Figure 5. Evolution of the lowest-lying Au10 isomers within a pool-GA run. Reproduced fromRef. [39] with permission from PCCP Owner Society.

Figure 6. Schematic flow chart of the BPGA program. The arrows show the local DFToptimizations. Reproduced from Ref. [40] with permission from The Royal Society ofChemistry.

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increases the GO efficiency.MEGAwas firstly applied for the GOof neutral andnegatively charged gas-phase Aun clusters (26 < n < 30) [130].

Current developments of the BPGA have the goal of making the GA moreversatile, to allow GO of more complex systems. This includes: interfacing itwith different QMpackages (allowing diverse electronic structuremethods); theexplicit consideration and optimization of cluster spin; and the optimization ofclusters in the presence of different chemical species (ligands or reactants).Further developments will include the simultaneous fitting of chemical andphysical properties during the GO process.

Other first principles approaches for optimizing cluster structures

In addition to the GA approach for the GO of clusters, several othereffective methods exist, which cannot be discussed in detail here. A briefintroduction is given below for a number of first principle-based GOtechniques which have been applied for cluster optimization.

Particle Swarm Optimization (PSO) is another nature-inspired algo-rithm, which is based on social-psychological behaviour of birds flockingand belongs to the category of swarm intelligence methods [140]. PSO is apopulation-based algorithm, in which the behaviour of every individual(agent) is determined by swarm intelligence to probe promising regions ofthe PES [54,141]. The GO of Li clusters was successfully carried out withPSO in combination with DFT within the CALYPSO methodology [142].

One of the most popular GO methods is the Basin Hopping (BH)algorithm, which is a Monte Carlo (MC) method that combines randomhopping moves with local minimization. Similar to a ‘Lamarckian’ GA, thePES is converted to a set of basins of attraction of all local minima [62,63].One of the first examples of a direct DFT-based BH GO of metal clusterwas carried out by Aprà et al. [143]. Usually the BH algorithm leads torandom jumps between nearby minima, which corresponds to an explora-tion of vicinity basins on the PES around the starting geometry. A dis-advantage of the traditional BH method is that there is a risk of beingtrapped in a very low local minimum (‘funnel’) if it is surrounded by highenergy barriers. To overcome this limitation, several methods have beendeveloped such as ‘jumping’ [144]. Other developments which usesdescriptors to tackle this trapping problem are the Parallel ExcitableWalkers (PEW) algorithm [145] and the population-based BH [112,146].The use of first principles methods such as DFT for the local minimizationstep with the BH algorithm has become a well-established method in theGO of clusters [147–153].

Another search scheme for GO is the threshold algorithm, which usesMC methods to explore the PES. The threshold algorithm performs astochastic investigation of the PES with the restriction to keep the energy

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below some threshold (lid) values [154,155]. The procedure can be con-sidered as constrained random walk, which allows an ergodic sampling ofthe available PES. The threshold algorithm is also suitable to investigateenergy barriers between local minima. New regions of the landscapebecome available by increasing the threshold and repeating the stochasticexploration of the PES. For example, DFT has been used in a combinedthreshold algorithm and GA study to investigate the structures and inter-conversions of low-lying isomers of Cu4Ag4 [156].

The minima-hopping-algorithm (MHA) uses molecular dynamics (MD)steps to explore the configurational space of the cluster system. The MHA ofGoedecker makes use of MD steps with subsequent local minimization and ahistorical list of the already visited minima to restrict the algorithm to searchnew regions of the PES [157]. MHA has been successfully used at the DFTlevel to investigate structures of Mg clusters [158].

Probably the first published GO method for cluster structures was simu-lated annealing (SA), but standard SA [56] is significantly outperformed byGA and BH methods [41–43]. The SA method consists of two steps: first theinitial system is heated up to a high temperature, with the system evolvingusing MC or MD simulations; subsequently, the temperature is slowlydecreased in a stepwise fashion. In order to perform first principles MDsimulations, SA was used in combination with DFT-based Car-Parrinellosimulations [159] and applied to predict the geometry of Se (Se3 to Se8) andsulphur clusters (S2 to S13) [160,161]. SA has also been coupled with HFcalculations, with subsequent random quenching, to search for the GM of LiFclusters [162].

Example applications of first principles GO methods to metal clustersand nanoalloys

In the following section we describe several examples of GO studies ofmetal clusters and nanoalloys employing first principles methods. For amore complete and detailed summary, the reader is referred to thefollowing references [10,47,53,163].

Au clusters

Clusters of noble metal elements are of considerable research interest,due to their intriguing optical properties and promising applications incatalysis and other technologies. Gold shows unique physical and che-mical properties, which are significantly influenced by relativistic effects[11,29,88,90,92,96,103,107,109,111]. There have been several GO studieson gold clusters using EPs followed by DFT relaxation [164–166].Assadollahzadeh and Schwerdtfeger performed a GO of neutral Aun (n

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= 2–20) clusters using a seeded GA-DFT approach. They predictedplanar structures as the putative GM up to Au10 and a tetrahedralstructure (Td) for Au20 [167]. The experimentally confirmed (by far-infrared-multiple photon dissociation (FIR-MPD) spectroscopy) [168]highly symmetric tetrahedral structure (Td) of Au20 was also obtainedwith a BH-DFT approach [143] and with the pool-BPGA at the DFTlevel by Shayeghi et al., who also found the same planar geometry forthe GM of Au10 [39]. The BH-DFT(GGA-PBE) method was applied byJiang and Walter for Au40, for which a twisted pyramid (C1 symmetry),with a tetrahedral core, was obtained as the putative GM [116]. NewGM structures, with core-shell structures, were proposed for Aun (n =27–30) using MEGA-DFT(GGA-PBE) [130].

For anionic Aun– (n = 36, 37, 38) clusters a BH-DFT approach was used

by Zeng et al. in a joint experimental (photoelectron spectroscopy) andtheoretical study [169]. They found for the most stable geometries core-shell structures with a tetrahedral core. A combined experimental(trapped-ion electron diffraction: TIED) and theoretical study, using DFTMD (TPSS meta-GGA) with a quenching algorithm, of Au12

– found quasi-isoenergetic 2D and 3D structures within the limits of the employedexperimental and theoretical methods [170].

The structure of small Au4+ clusters was elucidated by a combined

experimental (photodissociation spectroscopy) and theoretical approach,using the BCGA-DFT method [103]. A rhombic D2h structure wasobtained in accordance with the previous study of Gilb et al. [171].Moreover, the experimental depletion data indicates also an additionalcontribution of a Y-shaped isomer with C2v symmetry, which was alsoexperimentally observed by Dopfer et al. in a photodissociation experi-ment [172].

Ag clusters

Several joint experimental and theoretical studies have proven to be verypowerful tools in the structure determination of small silver cluster cations[49,102,103]. For Ag10

+, Ag8+, Ag6

+ and Ag4+, using the GA-DFT

approach, it was possible to explain experimental UV-Vis photodissocia-tion spectra in terms of one or more lowest lying energy isomers[49,102,103]. The GM structure of Ag4

+ is a rhombus with D2h symmetry,while the putative GM geometry of Ag6

+ is a bicapped tetrahedron (C2v)and the lowest lying isomer of Ag8

+ was found to be a pentagonalbipyramid with a single face capped (Cs). The putative Ag10

+ GM structureis built up of two interpenetrating pentagonal bipyramids (D2h).

The GO of neutral Agn (n = 5–12) clusters was carried out using theGA-DFT approach and compared to neutral gold and mixed AuAg clusters

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of the same size [173]. For neutral silver clusters, the 2D–3D transition wasdetermined to lie between the hexamer and heptamer clusters, i.e. signifi-cantly smaller than for gold.

Pt clusters

Pure platinum clusters (trimers to hexamers) as well as Ti- and V-doped Ptclusters have been investigated by Jennings and Johnston with the GA-DFT approach [174]. It was found that varying the spin multiplicity has astrong effect on pure Pt clusters regarding energies and structures, whereasdifferent effects are obtained for the doped ones, where partial spinquenching was observed. In general, higher spin states of PtTi clusterslead to higher energies. For singly V-doped Pt clusters the GM structuresare all doublets whereas the doubly doped clusters have triplet groundstates.

Ir clusters

Direct GO with spin polarized DFT was performed with the BPGA for Irn(n = 10–20) clusters [40]. In agreement with previous CCSD(T) andCASSCF calculations [175], simple cubic structures (or derivatives) werefound as the GM.

Sn clusters

There have been several first principle GO studies of main group metalclusters. For example, neutral Snn (n = 6–20) clusters were examined withthe DFT-GA approach in combination with electric beam deflection experi-ments [176]. The GM for n = 18–20 were found to be stacked prolatestructures, with at least one trigonal prism motif. Singlet states were foundto be most stable. Ahlrichs et al. examined the structures of tin cluster cationsSn3

+ to Sn15+, combining ion mobility measurements and unbiased GA-DFT

GO [177]. All the small Snn+ clusters up to the heptamer exhibit a Jahn-Teller

distortion. The GM structures for the larger clusters are mainly single ormultiple capped prisms or antiprisms. Schooss et al. employed a joint GA-DFT approach with TIED and collision-induced dissociation to determinethe structures of medium sized tin cluster anions Snn

− (n = 16–19) [178].They found prolate cluster geometries, which are composed of at least one ofthree frequently occurring building blocks: pentagonal bipyramid; tricappedtrigonal prism; bicapped tetragonal antiprism.

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Pb clusters

Lead is a heavier homologue of tin. Its properties are strongly influencedby relativistic spin-orbit (SO) effects. Götz et al. elucidated the structure ofneutral Pbn (n = 7–18), combining BCGA-DFT and molecular beamdeflection studies [101]. The GM cluster geometries were mainly sphericaland showed a strongly deviating growth pattern compared to tin clusters:whereas tin cluster adopts mainly prolate geometries, lead prefers a denserpacking and forms more spherical structures. In a later study, Götz et al.investigated the structure of larger neutral Pbn (n = 19–25, 31, 36, 54)clusters with BPGA-DFT, finding that Pbn clusters with up to 36 atoms arenot metallic [50]. They found oblate structures for the GM of Pb19 andPb20, while prolate structures were obtained for Pb21 to Pb25.

Al clusters

A GM search for Aln clusters (n = 2–30) was performed with a BH-MC-DFT algorithm [126]. The 2D–3D transition was found to take placebetween n = 4 and n = 5, with larger Al clusters exhibiting closely packedstructures. Al14 to Al19 display structures based on an underlying icosahe-dral Al13 motif.

Nanoalloys

Mixing two or more metals in order to form nanoalloys leads to atremendous increase in the tuning range of chemical and physicalproperties by adding two variables: composition and chemical ordering(the degree of mixing or segregation) [14]. However, as mentionedpreviously, this makes the GO procedure more complex.

Sn-Bi clusters

In a joint theoretical and experimental (electric field deflection method)approach, Snm-nBin clusters (m = 5–13, n = 1–2) were studied using theBCGA-DFT approach [100]. GO calculations used plane wave DFT. Lowenergy structures were reminimized using orbital-based DFT and electricdipole moments and polarizabilities were calculated for comparison withthe experiments. In nearly all cases, the lowest lying, metastable isomerswere found to be homotops of the GM. The low energy structures aretriangular faced polyhedral (‘deltahedra’), which are also found for boraneclusters and isoelectronic Zintl anions [179].

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Au-Ag, Au-Cu and Ag-Cu clusters

Nanoalloys of the coinage metals (Cu, Ag and Au) have been studied widely[14]. Heiles et al. employed BCGA-DFT to study the composition-depen-dence of the 2D–3D transition in neutral octameric gold–silver clusters (Au8-nAgn, n = 0–8) [98]. They predicted the 2D–3D transition to lie betweenAu6Ag2 and Au5Ag3. Subsequently, Heard et al. also used BCGA-DFT to findthe GM for neutral and charged Au8-nAgn (with n = 0–8) clusters [180]. Theyfound that the 2D–3D transition is highly sensitive to both charge andcomposition, with 3D structures becoming favoured at Au8

+ and Au2Ag6−

for cationic and anionic clusters, respectively, which is consistent with pre-vious studies that have shown that, for pure gold clusters, the 2D–3Dtransition occurs at smaller sizes for cations and larger sizes for anions,compared to neutral clusters [98,180].

The structure of mixed gold silver tetramers was elucidated by acombined experimental and theoretical approach. Using photodissocia-tion spectroscopy and BCGA-DFT [51], Zhao et al. performed anunbiased GM search of all possible compositions of AunAgm in thesize range 4 < n + m < 13 using GA-DFT(GGA-PBE) [173]. In agree-ment with previous studies, they found the same (octameric) GM struc-tures and ascertained that planar binary Au-Ag clusters retain thegeometry of the pure gold clusters, while 3D structures show a similarshape to the pure silver clusters.

The BCGA-DFT approach was used for the GM search of octamericAuCu and AgCu clusters [181]. All clusters except Au8 and AuCu7 werefound to have compact 3D structures. Thus, the 2D–3D transition takesplace after AuCu7.

Au-M clusters (M = Ir, Pd, Rh, Al, Bi)

In general, Au and Au-M Clusters (Au containing nanoalloys) have gainedmuch attention as catalysts for several chemical reactions [182-199]. Forexample, supported AuPd nanoparticles have been used as catalysts for theoxidation of carbon monoxide (CO) [195,200].

Au-Ir clusters

Due to the promise of Ir-doped Au nanoparticles in catalysis (CO oxida-tion), the GM search of AuIr nanoparticles is of particular importance.Adding Ir to Au catalysts leads to an improvement of the catalytic activityand prevents sintering [201,202]. The GO of AunIr4-n, AunIr5-n and AunIr6-n clusters was performed with the BPGA, using spin-polarized DFT [138].

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The GM structures for AunIr4-n, AunIr5-n and AunIr6-n were found to beplanar.

Au-Pd clusters

In a combined theoretical and experimental study (UV-vis depletionspectroscopy), the effect of doping Pd atoms into small cationic goldclusters (tetramer and pentamer) was investigated using the BCGA-DFTapproach [203]. It was found that Pd doping leads to the formation of 3Dstructures. Johnston et al. used the BPGA-DFT approach to perform asystematic search for the GM of neutral AunPdm clusters (n + m = 4–10)[204], finding that Pd-rich clusters tend to adopt 3D geometries, whereas2D geometries are only obtained for gold clusters doped with at most onePd atom. In a subsequent investigation, medium-sized Au-Pd clusters with11–18 atoms were globally optimized employing the BPGA with spinpolarized DFT [205].

Au-Rh clusters

GO of small AumRhn (3 < m + n < 7) clusters was performed usingBPGA-DFT [206]. Subsequently, MEGA-DFT calculation was used toinvestigate the GM structures of AumRhn (5 < m + n < 11) clusters[207]. Systems with a high gold concentration tend to form planarstructures with Rh atoms in highly coordinated positions and electrontransfer takes place from Rh to Au atoms. Clusters with a high concen-tration of Rh atoms form 3D structures, having an inner Rh coresurrounded by Au atoms.

Au-Al clusters

Anionic AunAlm– clusters with (n + m = 7, 8) were studied in a joint

approach consisting of experimental photoelectron spectroscopy and BH-DFT optimization [208]. The square bi-pyramidal structural motive of Al6

−

was preferably formed in AunAlm− clusters, which was explained in terms

of the dominance of the stronger Al–Al interactions compared to Au–Au.

Au-Bi clusters

In recent studies, Wang et al. investigated low-lying isomers of anionicAuBin

− (n = 4–8) clusters by combing BH-DFT GO and photoelectronspectroscopy [209]. The energies of the lowest energy DFT isomers werethen recalculated at the ab initio CCSD(T) level of theory. Mostly 3D

1096 M. JÄGER ET AL.

structures were found, with gold at higher coordination locations insidethe cluster, and charge transfer from Bi to Au was observed.

Pd-Ir clusters

Noble metals, such as Pd, Pt, Ir and Ru and their alloys, are promisingmaterials for chemical catalysis. GO studies of noble metal nanoalloys playan important role in the design of nanocatalysts for real-world applica-tions. BCGA-DFT was used to explore the PES of PdnIrN-n (n = 8–10)clusters [210]. For Ir-rich clusters, cube-based structures are preferred,with more close-packed structures found for Pd-rich clusters. Due to thesignificantly stronger Ir–Ir bonds, there is strong tendency for Pd–Irsegregation.

Pd-Co clusters

The GO of small PdnCom (2 < n + m < 8) nanoalloy clusters wasinvestigated using BCGA-DFT level [211]. Segregation of Pd atoms toperipheral positions in the cluster was observed, which results in a lowersurface energy and maximization of strong Co–Co bonds. Pd-inducedquenching of the magnetism was also noted.

Pt-Ru clusters

Subnanometre PtnRum (2 < n + m < 9) clusters were studied using BPGA-DFT [212]. It was noted that Pt atoms prefer peripheral sites, while Ruatoms favour central sites, with maximization of the stronger Ru–Rubonds. The putative GMs are depicted in Figure 7. The authors predictedRu@Pt core-shell structures to be energetically favoured with increasingcluster size.

Conclusions and outlook

Recent progress in the GO of metal clusters and nanoalloys, applying firstprinciples methods, has been reviewed here, with the main focus on GA-DFT. The use of first principles methods and sophisticated GO algorithmsis absolutely necessary to determine the GM structure for very smallclusters, because these ultra-small chemical particles tend to form unusualstructural motifs, which one would not expect based on chemical intuition.The challenge is locating the GM structure scales with system size andsystem complexity (bi- or multimetallic, ligated, surface-supported etc.).Hence, the development of more versatile and efficient GO algorithms isessential. For example, the efficiency of GO methods can be increased by

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combining the advantages of different algorithms or combining GO algo-rithms with powerful machine-learning tools. The development of com-puter hardware will make a major contribution to the investigation ofmore complex systems.

In conclusion, the GO of clusters is a very important task for both basicresearch and industry, since it can be used to predict experimental struc-tures and can lead to a deeper understanding of experimental data. GO ofnanosystems can also be employed for accurate prediction of particle shapeand properties and the design of new, improved catalysts, devices, etc. forreal-world applications.

Acknowledgements

RS and RLJ are grateful to their past and present co-workers and collaborators, whosework is cited herein, especially: Drs Sven Heiles and Armin Shayeghi (formerly of T. U.Darmstadt); Drs Andy Logsdail, Chris Heard and Jack Davis (formerly of the University ofBirmingham); Prof. Marcela Beltran and Drs Jorge Vargas and Fernando Buendia(UNAM, Mexico). RS acknowledges financial support from the DeutscheForschungsgemeinschaft (grant SCHA 885/15-1). MJ acknowledges a scholarship of theMerck’sche Gesellschaft für Kunst und Wissenschaft e.V. RLJ is grateful to the EPSRC (U.K.) for the Critical Mass Grant (EP/J010804/1) ‘TOUCAN: TOwards an Understanding ofCAtalysis on Nanoalloys’. Calculations were performed on the following HPC facilities: theMidPlus Regional Centre of Excellence for Computational Science, Engineering and

Figure 7. Global minima structures for PtnRum (2 < n + m < 9) clusters. Reproduced from Ref.[212] with permission from ACS Publications.

1098 M. JÄGER ET AL.

Mathematics, under EPSRC Grant EP/K000128/1; the University of Birmingham’sBlueBEAR HPC service, which provides a High-Performance Computing service to theUniversity’s research community; and the U.K.’s national HPC facility, ARCHER, both viamembership of the U.K.’s HPC Materials Chemistry Consortium, which is funded byEPSRC (EP/L000202), and via the TOUCAN grant.

Funding

RS acknowledges financial support from the Deutsche Forschungsgemeinschaft (grantSCHA 885/15-1). MJ acknowledges a scholarship of the Merck’sche Gesellschaft fürKunst und Wissenschaft e. V. RLJ is grateful to the EPSRC (U.K.) for the Critical MassGrant (EP/J010804/1) ‘TOUCAN: TOwards an Understanding of CAtalysis onNanoalloys’.

References

[1] L.N. Vicente and P.H. Calamai, J. Glob. Optim. 5 (1994) p. 291. doi:10.1007/BF01096458

[2] A. Khare and S. Rangnekar, Appl. Soft Comput. J. 13 (2013) p. 2997. doi:10.1016/j.asoc.2012.11.033

[3] M.C. Fu, F.W. Glover and J. April, Proc. 2005 Winter Simul. Conf. 1 (2005) p. 13.[4] G.G. Wang and S. Shan, J. Mech. Des. 129 (2007) p. 370. doi:10.1115/1.2429697[5] C.A. Floudas, I.G. Akrotirianakis, S. Caratzoulas, C.A. Meyer and J. Kallrath,

Comput. Chem. Eng. 29 (2005) p. 1185. doi:10.1016/j.compchemeng.2005.02.006[6] S. Andradóttir, Proc. 1998 Winter Simul. Conf. 1 (1998) p. 151.[7] M. Barbati, G. Bruno and A. Genovese, Expert Syst. Appl. 39 (2012) p. 6020.

doi:10.1016/j.eswa.2011.12.015[8] R. Dekker, Reliab. Eng. Syst. Saf. 51 (1996) p. 229. doi:10.1016/0951-8320(95)00076-

3[9] C.A. Floudas and C.E. Gounaris, J. Glob. Optim. 45 (2009) p. 3. doi:10.1007/s10898-

008-9332-8[10] S. Heiles and R.L. Johnston, Int. J. Quantum Chem. 113 (2013) p. 2091. doi:10.1002/

qua.24462[11] W. Eberhardt, Surf. Sci. 500 (2002) p. 242. doi:10.1016/S0039-6028(01)01564-3[12] P.K. Jain, K.S. Lee, I.H. El-Sayed and M.A. El-Sayed, J. Phys. Chem. B. 110 (2006) p.

7238. doi:10.1021/jp057170o[13] G. Schmid, Chem. Soc. Rev. 37 (2008) p. 1909. doi:10.1039/b713631p[14] R. Ferrando, J. Jellinek and R.L. Johnston, Chem. Rev. 108 (2008) p. 845.

doi:10.1021/cr040090g[15] H. Fan, P. Ju and S. Ai, Sensors Actuators B Chem. 149 (2010) p. 98. doi:10.1016/j.

snb.2010.06.023[16] D. Beydoun, R. Amal, G. Low and S. McEvoy, J. Nanoparticle Res. 1 (1999) p. 439.

doi:10.1023/A:1010044830871[17] O. Chen, J. Zhao, V.P. Chauhan, J. Cui, C. Wong, D.K. Harris, H. Wei, H.-S.

Han, D. Fukumura, R.K. Jain and M.G. Bawendi, Nat. Mater. 12 (2013) p. 445.doi:10.1038/nmat3539

[18] G.A. Ferguson, F. Mehmood, R.B. Rankin, J.P. Greeley, S. Vajda and L.A. Curtiss,Top. Catal. 55 (2012) p. 353. doi:10.1007/s11244-012-9804-4

ADVANCES IN PHYSICS: X 1099

[19] J. Li and J.Z. Zhang, Coord. Chem. Rev. 253 (2009) p. 3015. doi:10.1016/j.ccr.2009.07.017

[20] F.R. Negreiros, E. Aprà, G. Barcaro, L. Sementa, S. Vajda and A. Fortunelli,Nanoscale. 4 (2012) p. 1208. doi:10.1039/C1NR11051A

[21] A. Primo, A. Corma and H. García, Phys. Chem. Chem. Phys. 13 (2011) p. 886.doi:10.1039/c1cp21236b

[22] U. Landman, B. Yoon, C. Zhang, U. Heiz and M. Arenz, Top. Catal. 44 (2007) p.145. doi:10.1007/s11244-007-0288-6

[23] K. Saha, S.S. Agasti, C. Kim, X. Li and V.M. Rotello, Chem. Rev. 112 (2012) p. 2739.doi:10.1021/cr2001178

[24] G. Schmid, M. Bäumle, M. Geerkens, I. Heim, C. Osemann and T. Sawitowski,Chem. Soc. Rev. 28 (1999) p. 179. doi:10.1039/a801153b

[25] S. Linic, P. Christopher and D.B. Ingram, Nat. Mater. 10 (2011) p. 911. doi:10.1038/nmat3084

[26] P.V. Kamat, J. Phys. Chem. B. 106 (2002) p. 7729. doi:10.1021/jp0209289[27] M. Westphalen, U. Kreibig and J. Rostalski, Sol. Energy Mater. Sol. Cells. 61 (2000)

p. 97. doi:10.1016/S0927-0248(99)00100-2[28] G. Schön and U. Simon, Colloid Polym. Sci. 273 (1995) p. 101. doi:10.1007/

BF00654007[29] I. Tokareva, S. Minko, J.H. Fendler and E. Hutter, J. Am. Chem. Soc. 126 (2004) p.

15950. doi:10.1021/ja047044i[30] M. Daniel and D. Astruc, Chem. Rev. 104 (2004) p. 293.[31] U. Simon, G. Schön and G. Schmid, Angew. Chemie Int. Ed. Engl. 32 (1993) p. 250.

doi:10.1002/anie.199302501[32] R.M. Mohamed, D.L. McKinney and W.M. Sigmund, Mater. Sci. Eng. R Reports. 73

(2012) p. 1. doi:10.1016/j.mser.2011.09.001[33] P.V. Kamat, J. Phys. Chem. C. 112 (2008) p. 18737. doi:10.1021/jp806791s[34] J.L. West and N.J. Halas, Annu. Rev. Biomed. Eng. 5 (2003) p. 285. doi:10.1146/

annurev.bioeng.5.011303.120723[35] T.-H. Kim, K.-S. Cho, E.K. Lee, S.J. Lee, J. Chae, J.W. Kim, D.H. Kim, J.-Y. Kwon, G.

Amaratunga, S.Y. Lee, B.L. Choi, Y. Kuk, J.M. Kim and K. Kim, Nat. Photonics. 5(2011) p. 176. doi:10.1038/nphoton.2011.12

[36] A.A. Frost, J. Chem. Phys. 47 (1967) p. 3707. doi:10.1063/1.1701524[37] J. Simons, J. Phys. Chem. 95 (1991) p. 1017. doi:10.1021/j100156a002[38] D.M. Deaven, N. Tit, J.R. Morris and K.M. Ho, Chem. Phys. Lett. 256 (1996) p. 195.

doi:10.1016/0009-2614(96)00406-X[39] A. Shayeghi, D. Götz, J.B.A. Davis, R. Schäfer and R.L. Johnston, Phys. Chem.

Chem. Phys. 17 (2015) p. 2104. doi:10.1039/C4CP04323E[40] J.B.A. Davis, A. Shayeghi, S.L. Horswell and R.L. Johnston, Nanoscale. 7 (2015) p.

14032. doi:10.1039/C5NR03774C[41] B. Hartke, J. Phys. Chem. 97 (1993) p. 9973. doi:10.1021/j100141a013[42] D.M. Deaven and K.M. Ho, Phys. Rev. Lett. 75 (1995) p. 288. doi:10.1103/

PhysRevLett.75.1711[43] Y. Zeiri, Phys. Rev. E. 51 (1995) p. 2769. doi:10.1103/PhysRevE.51.R2769[44] J.A. Niesse and H.R. Mayne, J. Comput. Chem. 18 (1997) p. 1233. doi:10.1002/(SICI)

1096-987X(19970715)18:9<1233::AID-JCC11>3.0.CO;2-6[45] D.J. Wales and H.A. Scheraga, Science. 285 (1999) p. 1368. doi:10.1126/

science.285.5432.1368[46] R.L. Johnston,Dalt. Trans. (2003) p. 4193. doi:10.1039/b305686d[47] B. Hartke, Struct. Bond. 110 (2004) p. 33.

1100 M. JÄGER ET AL.

[48] B. Hartke, Wiley Interdiscip. Rev. Comput. Mol. Sci. 1 (2011) p. 879. doi:10.1002/wcms.70

[49] A. Shayeghi, D.A. Götz, R.L. Johnston and R. Schäfer, Eur. Phys. J. D. 69 (2015) p.152. doi:10.1140/epjd/e2015-60188-2

[50] D.A. Götz, A. Shayeghi, R.L. Johnston, P. Schwerdtfeger and R. Schäfer, Nanoscale.8 (2016) p. 11153. doi:10.1039/C6NR02080A

[51] A. Shayeghi, C.J. Heard, R.L. Johnston and R. Schäfer, J. Chem. Phys. 140 (2014) p.054312. doi:10.1063/1.4863443

[52] S. Heiles, K. Hofmann, R.L. Johnston and R. Schäfer, Chempluschem. 77 (2012) p.532. doi:10.1002/cplu.201200085

[53] C.J. Heard, R.L. Johnston, M. Nguyen and K. Boggovarapu, Eds. Clust. Struct. Bond.React., Springer, Berlin, 2016. 1–32.

[54] S.T. Call, D.Y. Zubarev and A.I. Bolodryev, J. Comput. Chem. 28 (2007) p. 1177.doi:10.1002/jcc.20688

[55] J. Zhang and M. Dolg, Phys. Chem. Chem. Phys. 17 (2015) p. 24173. doi:10.1039/C5CP04060D

[56] S. Kirkpatrick, C.D. Gelatt and M.P. Vecchi, Science. 220 (1983) p. 671. doi:10.1126/science.220.4595.360-d

[57] J.C. Schön, J. Phys. Condens. Matter. 8 (1996) p. 143. doi:10.1088/0953-8984/8/2/004

[58] S. Darby, T.V. Mortimer-Jones, R.L. Johnston and C. Roberts, J. Chem. Phys. 116(2002) p. 1536. doi:10.1063/1.1429658

[59] H. Kabrede and R. Hentschke, J. Phys. Chem. B. 107 (2003) p. 3914. doi:10.1021/jp027783q

[60] S. Li, R.L. Johnston and J.N. Murrell, J. Chem. Soc. Faraday Trans. 88 (1992) p.1229. doi:10.1039/ft9928801229

[61] B. Hartke, Chem. Phys. Lett. 258 (1996) p. 144. doi:10.1016/0009-2614(96)00629-X[62] D.J. Wales and J.P.K. Doye, J. Phys. Chem. A. 101 (1997) p. 5111. doi:10.1021/

jp970984n[63] J.P.K. Doye and D.J. Wales, New J. Chem. 1 (1998) p. 733. doi:10.1039/a709249k[64] S. Yoo and X.C. Zeng, J. Chem. Phys. 119 (2003) p. 1442. doi:10.1063/1.1581849[65] C. Roberts, R.L. Johnston and N.T. Wilson, Theor. Chem. Acc. 104 (2000) p. 123.

doi:10.1007/s002140000117[66] L.D. Lloyd, R.L. Johnston, S. Salhi and N.T. Wilson, J. Mater. Chem. 14 (2004) p.

1691. doi:10.1039/B313811A[67] I.L. Garzón, K. Michaelian, M.R. Beltrán, A. Posada-Amarillas, P. Ordejón, E.

Artacho, D. Sánchez-Portal and J.M. Soler, Phys. Rev. Lett. 81 (1998) p. 1600.doi:10.1103/PhysRevLett.81.1600

[68] T.A. Journal and F. Station, Phys. Rev. B. 62 (2003) p. 3344.[69] B. Hartke, Angew. Chemie - Int. Ed. 41 (2002) p. 1468. doi:10.1002/1521-3773

(20020503)41:9<1468::AID-ANIE1468>3.0.CO;2-K[70] G. Rossi, A. Rapallo, C. Mottet, A. Fortunelli, F. Baletto and R. Fernando, Phys. Rev.

Lett. 93 (2004) p. 105503. doi:10.1103/PhysRevLett.93.105503[71] D. Bochicchio and R. Ferrando, Nano Lett. 10 (2010) p. 4211. doi:10.1021/nl1017157[72] K. Laasonen, E. Panizon, D. Bochicchio and R. Ferrando, J. Phys. Chem. C. 117

(2013) p. 26405. doi:10.1021/jp410379u[73] E. Panizon and R. Ferrando, Nanoscale. 8 (2016) p. 15911. doi:10.1039/

C6NR03560D[74] S. Lysgaard, J.S.G. Mýrdal, H.A. Hansen and T. Vegge, Phys. Chem. Chem. Phys. 17

(2015) p. 28270. doi:10.1039/C5CP00298B

ADVANCES IN PHYSICS: X 1101

[75] G. Barcaro, L. Sementa and A. Fortunelli, Phys. Chem. Chem. Phys. 16 (2014) p.24256. doi:10.1039/C4CP03745F

[76] J.-P. Palomares-Baez, E. Panizon and R. Ferrando, Nano Lett. 17 (2017) p. 5394.doi:10.1021/acs.nanolett.7b01994

[77] J.Q. Goh, J. Akola and R. Ferrando, J. Phys. Chem. C. 121 (2017) p. 10809.doi:10.1021/acs.jpcc.6b11958

[78] D.J. Wales and M.P. Hodges, Chem. Phys. Lett. 286 (1998) p. 65. doi:10.1016/S0009-2614(98)00065-7

[79] R. Ismail, R. Ferrando and R.L. Johnston, J. Phys. Chem. C. 117 (2013) p. 293.doi:10.1021/jp3093435

[80] D.T. Tran and R.L. Johnston, Proc. R. Soc. A Math. Phys. Eng. Sci. 467 (2011) p.2004. doi:10.1098/rspa.2010.0562

[81] G. Rossi, R. Ferrando, A. Rapallo, A. Fortunelli, B.C. Curley, L.D. Lloyd and R.L.Johnston, J. Chem. Phys. 122 (2005) p. 194309. doi:10.1063/1.1898224

[82] F. Pittaway, L.O. Paz-Borbón, R.L. Johnston, H. Arslan, R. Ferrando, C. Mottet, G.Barcaro and A. Fortunelli, J. Phys. Chem. C. 113 (2009) p. 9141. doi:10.1021/jp9006075

[83] J.O. Joswig and M. Springborg, Phys. Rev. B - Condens. Matter Mater. Phys. 68(2003) 0854080. doi:10.1103/PhysRevB.68.085408

[84] R. Ferrando, A. Fortunelli and R.L. Johnston, Phys. Chem. Chem. Phys. 10 (2008) p.640. doi:10.1039/b710310g

[85] W. Cai, N. Shao, X. Shao, Z. Pan and J. Mol, Struct. Theochem. 678 (2004) p. 113.doi:10.1016/j.theochem.2004.03.017

[86] L.D. Lloyd and R.L. Johnston, Chem. Phys. 236 (1998) p. 107. doi:10.1016/S0301-0104(98)00180-3

[87] X. Shao, X. Liu and W. Cai, J. Chem. Theory Comput. 1 (2005) p. 762. doi:10.1021/ct049865j

[88] C. Harding, V. Habibpour, S. Kunz, A.N.S. Farnbacher, U. Heiz, B. Yoon and U.Landman, J. Am. Chem. Soc. 131 (2009) p. 538. doi:10.1021/ja804893b

[89] S. Bhattacharya, S.V. Levchenko, L.M. Ghiringhelli and M. Scheffler, New J. Phys. 16(2014) p. 123016. doi:10.1088/1367-2630/16/12/123016

[90] J. LaFemina, Handbook of Surface Science Volume I Physical Structure,1st, Elsevier,Liverpool, 1996. p. 139–180.

[91] F. Li and J.S. Lannin, Physics and Chemistry of Finite Systems: From Clusters toCrystals,1st, Springer Science & Business Media, Richmond, 1992. 1341–1346.

[92] R. Pacheco-Contreras, J.O. Juárez-Sánchez, M. Dessens-Félix, F. Aguilera-Granja, A.Fortunelli and A. Posada-Amarillas, Comput. Mater. Sci. 141 (2018) p. 30.doi:10.1016/j.commatsci.2017.09.022

[93] D. Alamanova, V.G. Grigoryan and M. Springborg, J. Phys. Chem. C. 111 (2007) p.12577. doi:10.1021/jp0717342

[94] D. Bochicchio and R. Ferrando, Phys. Rev. B - Condens. Matter Mater. Phys. 87(2013) p. 165435. doi:10.1103/PhysRevB.87.165435

[95] P. Koskinen and V. Mäkinen, Comput. Mater. Sci. 47 (2009) p. 237. doi:10.1016/j.commatsci.2009.07.013

[96] J. Cuny, N. Tarrat, F. Spiegelman, A. Huguenot, M. Rapacioli and J. Phys, Condens.Matter. 30 (2018) p. 303001. doi:10.1088/1361-648X/aacd6c

[97] T.W. Yen, T.L. Lim, T.L. Yoon and S.K. Lai, Comput. Phys. Commun. 220 (2017) p.143. doi:10.1016/j.cpc.2017.07.002

[98] S. Heiles, A.J. Logsdail, R. Schäfer and R.L. Johnston, Nanoscale. 4 (2012) p. 1109.doi:10.1039/C1NR11053E

1102 M. JÄGER ET AL.

[99] D.A. Götz, S. Heiles, R.L. Johnston and R. Schäfer, J. Chem. Phys. 136 (2012) p.186101. doi:10.1063/1.4717708

[100] S. Heiles, R.L. Johnston and R. Schäfer, J. Phys. Chem. A. 116 (2012) p. 7756.doi:10.1021/jp304321u

[101] D.A. Götz, A. Shayeghi, R.L. Johnston, P. Schwerdtfeger and R. Schäfer, J. Chem.Phys. 140 (2014) p. 164313. doi:10.1063/1.4872369

[102] A. Shayeghi, R.L. Johnston and R. Schäfer, J. Chem. Phys. 141 (2014) p. 181104.doi:10.1063/1.4901109

[103] A. Shayeghi, R.L. Johnston and R. Schäfer, Phys. Chem. Chem. Phys. 15 (2013) p.19715. doi:10.1039/c3cp52160e

[104] R.L. Johnston, Appplications of Evolutionary Computation in Chemistry, Springer,Berlin, 2004. p. 1–175.

[105] L.D. Lloyd, R.L. Johnston and S. Salhi, J. Comput. Chem. 26 (2005) p. 1069.doi:10.1002/jcc.20247

[106] F.B. Pereira and J.M.C. Marques, Evol. Intell. 2 (2009) p. 121. doi:10.1007/s12065-009-0020-5

[107] J. Smith, Evol. Comput. 15 (2007) p. 29. doi:10.1162/evco.2007.15.1.29[108] M. Lozano, F. Herrera and J.R. Cano, Inf. Sci. (Ny). 178 (2008) p. 4421. doi:10.1016/

j.ins.2008.07.031[109] M. Sierka, Prog. Surf. Sci. 85 (2010) p. 398. doi:10.1016/j.progsurf.2010.07.004[110] A. Cassioli, M. Locatelli and F. Schoen, Comput. Optim. Appl. 45 (2010) p. 257.

doi:10.1007/s10589-008-9194-5[111] B. Hartke, Jounral Comput. Chem. 20 (1999) p. 1752. doi:10.1002/(SICI)1096-987X

(199912)20:16<1752::AID-JCC7>3.0.CO;2-0[112] A. Grosso, M. Locatelli and F. Schoen, Math. Program. 110 (2007) p. 373.

doi:10.1007/s10107-006-0006-3[113] Y. Xiao and D.E. Williams, Chem. Phys. Lett. 215 (1993) p. 17. doi:10.1016/0009-

2614(93)89256-H[114] J.M.C. Marques and F.B. Pereira, Chem. Phys. Lett. 485 (2010) p. 211. doi:10.1016/j.

cplett.2009.11.059[115] F.B. Pereira, J.M.C. Marques, T. Leitão, J. Tavares, P. Siarry and Z. Michalewicz, Eds.

Natural Computing Series - Advances in Metaheuristics for Hard Optimization,Springer, Berlin, 2007. 223–250.

[116] Z. Chen, X. Jiang, J. Li and S. Li, J. Chem. Phys. 138 (2013) p. 214303. doi:10.1063/1.4807091

[117] A. Tomasulo and M.V. Ramakrishna, J. Chem. Phys. 105 (1996) 10449. doi:10.1063/1.472928

[118] M. Iwamatsu, J. Chem. Phys. 112 (2000) p. 10976. doi:10.1063/1.481737[119] L.D. Lloyd, R.L. Johnston, C. Roberts and T.V. Mortimer-Jones, Chem. Phys. Chem. 3

(2002) p. 408. doi:10.1002/1439-7641(20020517)3:5<408::AID-CPHC408>3.0.CO;2-G[120] M. Lemes, L. Marim and A. Dal Pino, Phys. Rev. A. 66 (2002) p. 023203.

doi:10.1103/PhysRevA.66.023203[121] N.T. Wilson and R.L. Johnston, J. Mater. Chem. 12 (2002) p. 2913. doi:10.1039/

B204069G[122] R.A. Lordeiro, F.F. Guimarães, J.C. Belchior and R.L. Johnston, Int. J. Quantum

Chem. 95 (2003) p. 112. doi:10.1002/qua.10660[123] A.N. Alexandrova, A.I. Boldyrev, Y.-J. Fu, X. Yang, X.-B. Wang and L.-S. Wang, J.

Chem. Phys. 121 (2004) p. 5709. doi:10.1063/1.1783276[124] J.M. Dieterich and B. Hartke, Mol. Phys. 108 (2010) p. 279. doi:10.1080/

00268970903446756

ADVANCES IN PHYSICS: X 1103

[125] A.J. Logsdail, Z.Y. Li and R.L. Johnston, J. Comput. Chem. 33 (2012) p. 391.doi:10.1002/jcc.21976

[126] F. Weigend, J. Chem. Phys. 141 (2014) p. 134103. doi:10.1063/1.4896658[127] L.B. Vilhelmsen and B. Hammer, J. Chem. Phys. 141 (2014) p. 044711. doi:10.1063/

1.4886337[128] S.M.A. Cruz, J.M.C. Marques and F.B. Pereira, J. Chem. Phys. 145 (2016) p. 154109.

doi:10.1063/1.4954660[129] M.S. Jørgensen, M.N. Groves and B. Hammer, J. Chem. Theory Comput. 13 (2017)

p. 1486. doi:10.1021/acs.jctc.6b01119[130] J.A. Vargas, F. Buendía and M.R. Beltrán, J. Phys. Chem. C. 121 (2017) p. 10982.

doi:10.1021/acs.jpcc.6b12848[131] T.C. Le and D.A. Winkler, Chem. Rev. 116 (2016) p. 6107. doi:10.1021/acs.

chemrev.5b00691[132] A.N. Alexandrova and A.I. Boldyrev, J. Chem. Theory Comput. 1 (2005) p. 566.

doi:10.1021/ct050093g[133] F.T. Silva, B.R.L. Galvão, G.P. Voga, M.X. Silva, D.D.C. Rodrigues and J.C. Belchior,

Chem. Phys. Lett. 639 (2015) p. 135. doi:10.1016/j.cplett.2015.09.016[134] G. Kresse and J. Hafner, Phys. Rev. B. 49 (1994) p. 14251. doi:10.1103/

PhysRevB.49.14251[135] G. Kresse and J. Hafner, Phys. Rev. B. 47 (1993) p. 558. doi:10.1103/

PhysRevB.47.558[136] G. Kresse and J. Furthmüller, Comput. Mater. Sci 6 (1996) p. 15. doi:10.1016/0927-

0256(96)00008-0[137] G. Kresse and J. Furthmüller, Phys. Rev. B. 54 (1996) p. 11169. doi:10.1103/

PhysRevB.54.11169[138] J.B.A. Davis, S.L. Horswell and R.L. Johnston, J. Phys. Chem. C. 120 (2016) p. 3759.

doi:10.1021/acs.jpcc.5b10226[139] C.J. Heard, S. Heiles, S. Vajda and R.L. Johnston, Nanoscale. 6 (2014) p. 11777.

doi:10.1039/C4NR03363A[140] R. Eberhart and J. Kennedy, Proc. 1995 IEEE Int. Conf. Neural Networks. 4 (1995) p. 39.[141] Y. Wang, J. Lv, L. Zhu, S. Lu, K. Yin, Q. Li, H. Wang, L. Zhang and Y. Ma, J. Phys.

Condens. Matter. 27 (2015) p. 203203. doi:10.1088/0953-8984/27/20/203203[142] J. Lv, Y. Wang, L. Zhu and Y. Ma, J. Chem. Phys. 137 (2012) p. 084104. doi:10.1063/

1.4746757[143] E. Aprà, R. Ferrando and A. Fortunelli, Phys. Rev. B - Condens. Matter Mater. Phys.

73 (2006) p. 205414. doi:10.1103/PhysRevB.73.205414[144] M. Iwamatsu and Y. Okabe, Chem. Phys. Lett. 399 (2004) p. 396. doi:10.1016/j.

cplett.2004.10.032[145] G. Rossi and R. Ferrando, Chem. Phys. Lett. 423 (2006) p. 17. doi:10.1016/j.

cplett.2006.03.003[146] W. Pullan, J. Comput. Chem. 26 (2005) p. 899. doi:10.1002/jcc.20226[147] A. Bruma, F.R. Negreiros, S. Xie, T. Tsukuda, R.L. Johnston, A. Fortunelli and Z.Y.

Li, Nanoscale. 5 (2013) p. 9620. doi:10.1039/c3nr01852k[148] R. Gehrke and K. Reuter, Phys. Rev. B - Condens. Matter Mater. Phys. 79 (2009) p.

085412. doi:10.1103/PhysRevB.79.085412[149] S. Jiang, Y.R. Liu, T. Huang, H. Wen, K.M. Xu, W.X. Zhao, W.J. Zhang and W.

Huang, J. Comput. Chem. 35 (2014) p. 159. doi:10.1002/jcc.23477[150] Y. Gao, S. Bulusu and X.C. Zeng, Chem. Phys. Chem. 7 (2006) p. 2275. doi:10.1002/

cphc.200600472[151] H. Do and N.A. Besley, J. Chem. Phys. 137 (2012) p. 134106. doi:10.1063/1.4755994

1104 M. JÄGER ET AL.

[152] G.G. Rondina and J.L.F. Da Silva, J. Chem. Inf. Model. 53 (2013) p. 2282.doi:10.1021/ci400224z

[153] R. Ouyang, Y. Xie and D. Jiang, Nanoscale. 7 (2015) p. 14817. doi:10.1039/C5NR03903G

[154] P. Sibani, J.C. Schön, P. Salamon and J.-O. Andersson, Eur. Lett. 22 (1993) p. 479.doi:10.1209/0295-5075/22/7/001

[155] S. Neelamraju, C. Oligschleger and J.C. Schön, J. Chem. Phys. 147 (2017) p. 152713.doi:10.1063/1.4985912

[156] C.J. Heard, R.L. Johnston and J.C. Schön, Chem. Phys. Chem. 16 (2015) p. 1461.doi:10.1002/cphc.201402887

[157] S. Goedecker, J. Chem. Phys. 120 (2004) p. 9911. doi:10.1063/1.1724816[158] I. Heidari, S. De, S.M. Ghazi, S. Goedecker and D.G. Kanhere, J. Phys. Chem. A. 115

(2011) p. 12307. doi:10.1021/jp204442e[159] R. Car and M. Parrinello, Phys. Rev. Lett. 55 (1985) p. 2471. doi:10.1103/

PhysRevLett.55.791[160] D. Hohl, R.O. Jones, R. Car and M. Parrinello, Chem. Phys. Lett. 139 (1987) p. 540.

doi:10.1016/0009-2614(87)87339-6[161] D. Hohl, R.O. Jones, R. Car and M. Parrinello, J. Chem. Phys. 89 (1988) p. 6823.

doi:10.1063/1.455356[162] K. Doll, J.C. Schön and M. Jansen, J. Chem. Phys. 133 (2010) p. 024107. doi:10.1063/

1.3455708[163] J. Zhao, X. Huang, R. Shi, L. Tang, Y. Su and L. Sai, Chem. Model. 12 (2016) p. 249.[164] B.S. De Bas, M.J. Ford, M.B. Cortie and J. Mol, Struct. ´Theochem. 686 (2004) p.

193. doi:10.1016/j.theochem.2004.08.029[165] J.M. Soler, I.L. Garzón and J.D. Joannopoulos, Solid State Commun. 117 (2001) p.

621. doi:10.1016/S0038-1098(00)00493-2[166] N.T. Wilson and R.L. Johnston, Eur. Phys. J. D. 12 (2000) p. 161. doi:10.1007/

s100530070053[167] B. Assadollahzadeh and P. Schwerdtfeger, J. Chem. Phys. 131 (2009) p. 064306.

doi:10.1063/1.3175013[168] P. Gruene, D.M. Rayner, B. Redlich, A.F.G. van der Meer, J.T. Lyon, G. Meijer and

A. Fielicke, Science. 321 (2008) p. 674. doi:10.1126/science.1161166[169] N. Shao, W. Huang, W.N. Mei, L.S. Wang, Q. Wu and X.C. Zeng, J. Phys. Chem. C.

118 (2014) p. 6887. doi:10.1021/jp500582t[170] M.P. Johansson, A. Lechtken, D. Schooss, M.M. Kappes and F. Furche, Phys. Rev. A

- At. Mol. Opt. Phys. 77 (2008) p. 053202. doi:10.1103/PhysRevA.77.053202[171] S. Gilb, P. Weis, F. Furche, R. Ahlrichs and M.M. Kappes, J. Chem. Phys. 116 (2002)

p. 4094. doi:10.1063/1.1445121[172] B.K.A. Jaeger, M. Savoca, O. Dopfer and N.X. Truong, Int. J. Mass Spectrom. 402

(2016) p. 49. doi:10.1016/j.ijms.2016.02.019[173] L. Hong, H. Wang, J. Cheng, X. Huang, L. Sai and J. Zhao, Comput. Theor. Chem.

993 (2012) p. 36. doi:10.1016/j.comptc.2012.05.027[174] P.C. Jennings and R.L. Johnston, Comput. Theor. Chem. 1021 (2013) p. 91.

doi:10.1016/j.comptc.2013.06.033[175] M. Chen and D.A. Dixon, J. Phys. Chem. A. 117 (2013) p. 3676. doi:10.1021/

jp311003d[176] S. Schäfer, B. Assadollahzadeh, M. Mehring, P. Schwerdtfeger and R. Schäfer, J.

Phys. Chem. A. 112 (2008) p. 12312. doi:10.1021/jp8030754[177] N. Drebov, E. Oger, T. Rapps, R. Kelting, D. Schooss, P. Weis, M.M. Kappes and R.

Ahlrichs, J. Chem. Phys. 133 (2010) p. 224302. doi:10.1063/1.3514907

ADVANCES IN PHYSICS: X 1105

[178] A. Wiesel, N. Drebov, T. Rapps, R. Ahlrichs, U. Schwarz, R. Kelting, P. Weis, M.M.Kappes and D. Schooss, Phys. Chem. Chem. Phys. 14 (2012) p. 234. doi:10.1039/C1CP22874A

[179] J.D. Corbett, Angew. Chemie Int. Ed. 39 (2000) p. 670. doi:10.1002/(SICI)1521-3773(20000218)39:4<670::AID-ANIE670>3.0.CO;2-M

[180] C. Heard, A. Shayeghi, R. Schäfer and R. Johnston, Z. Phys. Chem. 230 (2016) p.955. doi:10.1515/zpch-2015-0721

[181] C.J. Heard and R.L. Johnston, Eur. Phys. J. D. 67 (2013) p. 34. doi:10.1140/epjd/e2012-30601-7

[182] W. He, X. Wu, J. Liu, X. Hu, K. Zhang, S. Hou, W. Zhou and S. Xie, Chem. Mater.22 (2010) p. 2988. doi:10.1021/cm100393v

[183] J. Yan, X. Zhang, T. Akita, M. Haruta and Q. Xu, J. Am. Chem. Soc. 132 (2010) p.5326. doi:10.1021/ja910513h

[184] R.M. Finch, N.A. Hodge, G.J. Hutchings, A. Meagher, Q.A. Pankhurst, M. Rafiq, H.Siddiqui, F.E. Wagner and R. Whyman, Phys. Chem. Chem. Phys. 1 (1999) p. 485.doi:10.1039/a808208a

[185] J.K. Edwards, B. Solsona, P. Landon, A.F. Carley, A. Herzing, M. Watanabe, C.J.Kiely and G.J. Hutchings, J. Mater. Chem. 15 (2005) p. 4595. doi:10.1039/b509542e

[186] G.J. Hutchings, Chem. Commun. 7345 (2008) p. 1148. doi:10.1039/B712305C[187] G.J. Hutchings, J. Mater. Chem. 19 (2009) p. 1222. doi:10.1039/B812300B[188] M. Haruta, Chem. Rec. 3 (2003) p. 75. doi:10.1002/tcr.10053[189] A.S.K. Hashmi and G.J. Hutchings, Angew. Chemie - Int. Ed. 45 (2006) p. 7896.

doi:10.1002/anie.200602454[190] M.D. Hughes, Y.-J. Xu, P. Jenkins, P. McMorn, P. Landon, D.I. Enache, A.F. Carley,

G.A. Attard, G.J. Hutchings, F. King, E.H. Stitt, P. Johnston, K. Griffin and C.J.Kiely, Nature. 437 (2005) p. 1132. doi:10.1038/nature04190

[191] M. Sankar, E. Nowicka, R. Tiruvalam, Q. He, S.H. Taylor, C.J. Kiely, D. Bethell, D.W. Knight and G.J. Hutchings, Chem. - A Eur. J. 17 (2011) p. 6524. doi:10.1002/chem.201003484

[192] C.J. Zhong and M.M. Maye, Adv. Mater. 13 (2001) p. 1507.[193] G.J. Hutchings, Catal. Today. 100 (2005) p. 55. doi:10.1016/j.cattod.2004.12.016[194] G.J. Hutchings and M. Haruta, Appl. Catal. A Gen. 291 (2005) p. 2. doi:10.1016/j.

apcata.2005.05.044[195] J.A. Lopez-Sanchez, N. Dimitratos, N. Glanville, L. Kesavan, C. Hammond, J.K.

Edwards, A.F. Carley, C.J. Kiely and G.J. Hutchings, Appl. Catal. A Gen. 391 (2011)p. 400. doi:10.1016/j.apcata.2010.05.010

[196] G.J. Hutchings, Gold Bull. 29 (1996) p. 123. doi:10.1007/BF03214746[197] G.J. Hutchings, Gold Bull. 37 (2004) p. 3. doi:10.1007/BF03215511[198] D.I. Enache, J.K. Edwards, P. Landon, B. Solsona-Espriu, A.F. Carley, A.A. Herzing,

M. Watanabe, C.J. Kiely, D.W. Knight and G.J. Hutchings, Science. 311 (2006) p.362. doi:10.1126/science.1120560

[199] M. Haruta, Nature. 437 (2005) p. 1098. doi:10.1038/4371098a[200] L. Guczi, A. Beck, A. Horváth, Z. Koppány, G. Stefler, K. Frey, I. Sajó, O. Geszti, D.

Bazin, J. Lynch and J. Mol, Catal. A Chem. 204–205 (2003) p. 545. doi:10.1016/S1381-1169(03)00337-6

[201] X. Bokhimi, R. Zanella and C. Angeles-Chavez, J. Phys. Chem. C. 114 (2010) p.14101. doi:10.1021/jp103053e

[202] M. Saniger, G. Díaz, R. Zanella, H. Ramı, P. Santiago and A. Gómez-Cortés, J. Phys.Chem. C. 113 (2009) p. 9710. doi:10.1021/jp810905n

1106 M. JÄGER ET AL.

[203] V. Kaydashev, P. Ferrari, C. Heard, E. Janssens, R.L. Johnston and P. Lievens, Part.Part. Syst. Charact. 33 (2016) p. 364. doi:10.1002/ppsc.201600036

[204] H.A. Hussein, B.A. Davis and R.L. Johnston, Phys. Chem. Chem. Phys. 18 (2016) p.26133. doi:10.1039/c6cp01154c

[205] H.A. Hussein, I. Demiroglu and R.L. Johnston, Eur. Phys. J. B. 91 (2018) p. 34.doi:10.1140/epjb/e2017-80314-2

[206] F. Buendía, J.A. Vargas, M.R. Beltrán, J.B.A. Davis and R.L. Johnston, Phys. Chem.Chem. Phys. 18 (2016) p. 22122. doi:10.1039/c6cp01154c

[207] F. Buendía, J.A. Vargas, R.L. Johnston and M.R. Beltrán, Comput. Theor. Chem.1119 (2017) p. 51. doi:10.1016/j.comptc.2017.09.008

[208] N.S. Khetrapal, T. Jian, R. Pal, G.V. Lopez, S. Pande, L.-S. Wang and X.C. Zeng,Nanoscale. 8 (2016) p. 9805. doi:10.1039/C6NR01506A

[209] S. Pande, T. Jian, N.S. Khetrapal, L.S. Wang and X.C. Zeng, J. Phys. Chem. C. 122(2018) p. 6947. doi:10.1021/acs.jpcc.8b00166

[210] J.B.A. Davis, S.L. Horswell and R.L. Johnston, J. Phys. Chem. A. 118 (2014) p. 208.doi:10.1021/jp408519z

[211] M. Aslan, J.B.A. Davis and R.L. Johnston, Phys. Chem. Chem. Phys. 18 (2016) p.6676. doi:10.1039/c6cp01154c

[212] I. Demiroglu, K. Yao, H.A. Hussein and R.L. Johnston, J. Phys. Chem. C. 121 (2017)p. 10773. doi:10.1021/acs.jpcc.6b11329

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